Understanding the Mixing Patterns of Social Networks: The Impact of Cores, Link Directions, and Dynamics

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1 Understanding the Mixing Patterns of Socia Networks: The Impact of Cores, Link Directions, and Dynamics [Last revised on May 22, 2011] Abedeaziz Mohaisen Huy Tran Nichoas Hopper Yongdae Kim University of Minnesota Twin Cities ABSTRACT We extend upon our earier resuts measuring the mixing time and advance our understanding of the probem in severa directions. We reate the mixing time of socia graphs to graph degeneracy, which captures cohesiveness of the graph. We experimentay show that fast-mixing graphs tend to have a arger singe core whereas sow-mixing graphs tend to have smaer mutipe cores. Whie this study provides quantitative and empirica evidence reating the mixing time to core structure of socia graphs, it aso agrees with our previous remarks on the potentia reationship between the tight-knit community structure and sow mixing socia graphs. Equipped with with these findings, we advance this vein of research in severa directions. First, we study severa heuristics to improve the mixing time in sow-mixing graphs using their topoogica structures. Second, we study the impact of ink direction and ink dynamics on the quaity of the mixing time. As anticipated, we find that dynamic socia graphs have time-varying mixing characteristics. Counterintuitivey, we find directed graphs are faster-mixing than the same graphs when considered without edge direction. 1. INTRODUCTION Leveraging socia ties for buiding trustworthy computing services is becoming quite popuar, and promises many primitives and appications for communication and security. Such appications and primitives benefit from both the trust exhibited in the underying socia networks which rationaizes coaboration among nodes in the services buit on top of it and other agorithmic properties, which support the argument for the effectiveness of appications buit on top of these networks. Whie most appications and primitives buit on top of socia networks share a commonaity of purpose in using trust in the underying graphs (e.g., for ensuring coaboration, rationaizing assumptions about the nature of the underying socia graph and attackers capabiities, among many others), the agorithmic properties used in them differ greaty [30]. In the foowing, we eaborate on some of these expoited properties and their appications. For exampe, the betweenness of nodes in socia graphs which captures how we-situated a node is on the path between other nodes is used in [36] to buid a Sybi defense in mobie networks. The betweenness and simiarity are used in [6] for improving routing in deay toerant networks. The coseness and connectivity are used in [13] for efficient content sharing and anonymity. Connectivity and transitive trust are used in [26] for improving Internet routing. Another exampe of appications is socia network based Sybi defenses, where severa designs have been proposed to defend against the Sybi attack an attack in which a misbehaving user caims mutipe identities to infuence the system s behavior [8, 23, 29, 36, 40, 46, 50, 51]. In most of these designs and defenses, trust is used to rationaize the difficuty of penetrating the socia graph and estabishing arbitrariy many inks that thwart the utiity of the defense mechanism, whereas the fast-mixing property is used to argue for the effectiveness of the detection of Sybi identities, and to produce a feasibe soution. The same agorithmic property, of fast-mixing socia graphs, is used in a cosey reated direction for demonstrating the utiity of socia graphs as good mixers for potentia depoyment of anonymous communication networks on top of them [34]. The same property is used in mobie networks for fast content dissemination [27]. Despite the importance of these properties to the usabiity of such appications, ess effort is spent understanding the quaity of such properties in rea-word socia graphs, and even ess effort is spent reating these properties to other topoogica characteristics of socia graphs. Motivated by the necessity of the assumption to operate these defenses, we have recenty [33] measured the mixing time in severa rea-word socia and information graphs and estabished that many socia graphs are sower mixing than beieved in the iterature. We further demonstrated that the quaity of the mixing time required for operating Sybi defenses for the majority of honest nodes in the socia graph is ess than assumed in the iterature. At the down side, we have shown that some socia graphs despite the reaxation of the 1

2 required quaity of the mixing time have poor mixing characteristics which makes operating Sybi defenses on top of them infeasibe. 1.1 Main Contributions In this work, we extend upon our previous resuts and vasty advance our understanding of the mixing characteristics of socia graphs in severa directions, each of which is an interesting contribution in its own right. In the first direction, we re-examine the mixing characteristics of socia graphs and reate the mixing pattern in such graphs to their topoogica structure. In particuar, we reate the mixing time of socia graphs to graph degeneracy, which captures the cohesiveness of the graph. We show that fast-mixing graphs are cohesive and consist of a singe core even after the decomposition of the graph into its k-core [25] (a subgraph from the origina graph obtained by iterativey pruning node with degree ess than k). On the other hand, we show that sow-mixing socia graphs tend to be ess cohesive where they consist of mutipe cores after the decomposition of their graph to the k-core for reativey sma k. In the second direction, we investigate the use of graph structure to improve the mixing time of sowmixing graphs. Arguing that many of the existing soutions in the iterature for speeding up the mixing time are impractica to the context of many appications proposed on top of socia networks, we consider severa heuristics for improving the mixing time of sow-mixing graphs. Our heuristics are motivated by our findings on the structure of sow-mixing graphs and use auxiiary edges to wire separate cores. We investigate the potentia of these techniques on the improvement of the mixing time. Counterintuitivey, we show that the improvement of the mixing time does not depend on the number of edges used for connecting the separate cores but rather the way they are used: adding a sma number of edges between cores improves the mixing time whie adding more edges arbitrariy woud reduce this quaity. In the third direction, we investigate the mixing time of directed graphs. Despite that many socia graphs are directed by nature, they are converted to undirected graphs by competing their degrees [23, 50]. The main caim made in this context is that the majority of directed edges are in both directions between any pair of nodes, except in a sma portion of nodes that break the symmetry of edges. However, the extent to which the modification of graphs in that way woud affect the mixing time is not cear. Despite that the number of edges that break the symmetry of direction in edges coud be as high as 50%, we unvei a surprising and a counterintuitive resut by demonstrating that directed graphs are as good mixing as their undirected counterparts. In fact, we show two cases where undirected graphs are sower-mixing than the same graphs when considered with directions. Finay, we formaize the probem of measuring the mixing time in dynamic graphs from an appication point of view, and provide the sufficient mathematica toos to do that by extending upon the toos used for measuring static graphs. Using these toos, and two representative socia graphs of dynamic structure, we measure the mixing time of dynamic graphs. As anticipated, we show that the mixing time of dynamic graphs is time-dependent. We further quantitativey measure how the variation in the mixing time of dynamic graphs woud affect the operation of Sybi defenses. 1.2 Roadmap The rest of this paper is structured as foows. In section 2, we review reated work to motivate for our paper and mutifod contributions. In section 3 we outine the preiminaries by reviewing theoretica toos required for measuring the mixing time in simpe graphs, and show motivationa exampes on the mixing patterns of some rea-word graphs. In section 4 we expore the reationship between the mixing time of socia graphs and their core structures. In section 5 we expore methods for improving the mixing characteristics of sow-mixing socia graphs motivated by our findings on the structure of these graphs. In section 6 we resove the probem of direction in graphs by discussing toos for measuring the mixing time in a set of these graphs, and quantitivey compare that to undirected graphs. In section 7 we expore the mixing time of dynamic graphs and show measurements of these graphs. Finay, in section 8 we outine future work directions and draw concuding remarks. 2. RELATED WORK Using socia networks to buid primitives and services by expoiting socia structures, socia network properties, and trust in socia graphs has been a growing area of research, where severa directions are investigated to sove chaenging probems. Socia networks measurements is yet another active area of research, where most of the measured aspects in this paper have their share of interest in other contexts. Finay, improving the mixing characteristics of socia networks, and formaizing the mixing characteristics of random waks in severa graph settings have been activey investigated in the iterature. Here, we eaborate on the reevant prior work in the iterature and emphasize how it differs from our work. However, we imit ourseves to the domain of appications buit on top of socia networks for Sybi defenses. 2.1 Sybi Defenses Using Socia Networks Whie the idea of using socia networks to improve security in distributed systems has been investigated 2

3 in [26] and [7], the first forma attempt to use socia networks for defending attacks was in SybiGuard [51], which is uses fast-mixing graphs for Sybi detection. The same work has been extended and improved in SybiLimit [50]. Both SybiGuard and SybiLimit did not measure the mixing time, which is necessary for their operation. However, SybiLimit showed end-toend resuts demonstrating that rea-word socia graphs mix we enough to support its requirements. Aso using fast-mixing graphs, Danezis and Mitta introduced SybiInfer, an inference mechanism for Sybi nodes detection [8]. Tran et a. used the fast-mixing socia graphs, and a ticket distribution mechanism that benefits from the mixing characteristics of socia graphs, for buiding a Sybi resistant voting system caed SumUp [46]. Lesniewski-Laas et a. [23] used the same property to buid a Sybi-proof DHT system. Based on the expansion which is reated to the mixing time, Tran et a. introduced GateKeeper [45] which improves SybiLimit by reducing the number of sybi introduced per attack edge to O(1). Using transitive trust and impicit assumptions on mixing characteristics of socia graphs, Sirivianos et a. [40] and Misove et a. [29] introduced two fitering agorithms of unwanted communications, both of which try to mitigate the impact of Sybi identities. Though authors of most of these designs performed experiments to show insights on practicaity of their designs, none of them measured the mixing time or any other property used for their operation directy from socia graphs. Measuring these properties to understand the quaity required for the operation of these designs is possibe in most socia graphs. This work is dedicated for understanding the main used property, the mixing time, under different scenarios. 2.2 Defense Anaysis and Mixing Measurements The use of assumptions in buiding Sybi defenses on top of socia networks ike the fast-mixing without verification motivated for investigating whether such systems work on various socia graphs or not. Viswanath et a. [48] conducted an experimenta anaysis of Sybi defenses based on socia networks by comparing different defenses (SybiGuard [51], SybiLimit [50], Sybi- Infer [8], and SumUp [46]). They demonstrated that different Sybi defenses work by ranking different nodes based on how they are we-connected to a trusted node. Aso, they demonstrate that different Sybi defenses are sensitive to community structure in socia networks and argue that community detection agorithms can be used to repace the random wak based Sybi defenses. In [33], Mohaisen et a. measured the mixing time in severa socia graphs and demonstrated that socia graphs are sower mixing than beieved in iterature. Furthermore, they noticed that mixing patterns in socia graphs are associated with the underying socia mode, where socia networks with confined socia modes, and strict trust properties, are sow mixing whereas socia networks with ess strict trust properties are fast mixing. This observation is used in [30] to account for trust in socia network-based Sybi defenses using moduated random waks. Finay, Deamico et a. [9] measured the mixing time in four datasets (Epinion, OpenPGP, DBLP, and Advogato) using the samping method and the mode in (3). Their work was motivated by evauating the effectiveness of a wide range of designs based on both the mixing time and transitive trust. The authors caimed that rea-word socia networks meet theoretica assumptions of the Sybi defenses without showing that on any particuar defense. The main concusion made in [9] is that the mixing time is not associated with any of the known characteristics of the socia graphs. Our work in this paper is motivated by this part of reated work, where our work advances existing findings vasty in many directions. We reate the mixing time to graph structure, propose heuristics to improve the mixing time based on this structure, measure the mixing time for naturay directed graphs, and formaize and measure the mixing time for dynamic graphs. In each of these directions we show severa interesting findings based on our measurements. 2.3 Mixing and Graph Structure To the best of our knowedge, there has been no prior work in the iterature on understanding the mixing time of socia graphs and reating it to graph structure, though there has been severa works on understanding graph structura properties that coud be indirecty reated to the mixing time (such as diameter and custers) [21,22]. Thus our work is the first of its own type to consider this probem. On the other hand, controing the mixing characteristics of socia graphs, by either sowing them down or speeding them up, has been considered in severa papers incuding [3, 11, 18, 30, 37] In [37] it is shown that mutipe random waks can be used to speed up the mixing characteristics of socia graphs. In [11], a method is proposed for samping mutigraph, where it is caimed that these graphs are faster mixing than their simpe counterparts. In [18] a heuristic is used to sampe from graphs with respect to a certain biased distribution whie maintaining a fast convergence rate to that distribution. In [3] a method for speeding up convergence to stationary distribution by providing nodes with random uniform jumps capabiities to the entire graph which is in essence simiar to PageRank [35] for directed graphs is proposed and shown to speed up mixing characteristics of socia graphs. Finay, in [30] severa moduated random waks are proposed to contro the mixing time (mosty by re- 3

4 ducing it) by imiting waks to trusted inks, which in turn improves the operation of appications buit on top of these graphs. Whie these designs are of a great vaue, we argue that in many rea-word scenarios they cannot be used directy to contro the mixing time of socia graphs due to their high cost and high risk to the utiity of the intended appications by the improved mixing time. More detais are in section Properties of Dynamic and Directed Graphs Understanding dynamics in socia and information networks has been extensivey studied in the iterature [4, 12, 16, 17, 41 44, 47, 49]. For exampe, in [12] Gobeck studied patterns of dynamics of socia memberships and reationships. In [16], Keinberg studied cascading behavior in networks in genera, which appies to socia networks under dynamics as we. In [4], Backstrom et a. studied group formation under dynamics in socia networks. In [42], Tang et a. studied community evoution in dynamic socia networks. In [43], Tantipathananandh et a proposed a framework for community detection in dynamic socia graphs. In [49], Wison et a. studied interaction graphs by anayzing and comparing their structura properties, and measured the impication of their structure on appications, incuding Sybi defenses (SybiGuard in particuar). Whie the resuts show significant trend of dynamics, no measurements for how these dynamics quantitativey affect the mixing characteristic of the socia graph is provided. Simiar to that, Viswanath et a., in [47], considered the (voume-wise) evoution of interactions among users in Facebook, but did not measure the mixing time of the time-varying socia graph. In [30] we measured the mixing time of an interaction graph borrowed from [49] and compared it to the reationshipbased graph. We demonstrated that the former graph is sower mixing than the atter one, but did not consider the probem of measuring the mixing time of the evoving graph at different times due to the ack of data and toos. Findings in most of these works are different in essence. However, they convey simiar concusions by pointing out the impact of dynamics on measured characteristics in socia graphs. Yet, none of these works considered how these dynamics affect the mixing time. In this work, we go one step farther by formaizing timeevoving graphs, define their mixing time, and measure it in two rea-word traces. Measuring the mixing time in directed graph is not expored yet too, though there has been some recent theoretica efforts on formaizing the probem of fastmixing directed graphs, as in [15]. In this paper, as the first resut in that direction, we eaborate on formaizing the mixing time of directed graphs, show how to measure it, and measure it for severa directed graphs. We compare the measurements to undirected graphs derived by omitting graph directions. 3. THE MIXING TIME IN SIMPLE GRAPHS Here we eaborate on the method used in measuring the mixing time in undirected unweighted (simpe) graphs, which is simpest to consider. We begin by formaizing the mode of these graphs, the mixing time, and some motivationa measurements borrowed from our prior work in [33]. 3.1 Graph Mode and Uniform Waks Let G = (V, E) be a simpe undirected and unweighted graph, where V is the set of vertices of G such that V = n and E is the set of edges in G such that E = m. For G, we define A = [a ij ] n n as the adjacency matrix, where a ij = { 1 v i v j 0 otherwise. (1) (v i v j means that v i is adjacent to v j ). Let the number of nodes in V adjacent to v i be deg(v i ) (aso computed from A as deg(v i ) = j a ij). For G, we define the stochastic transition probabiity matrix P = [p ij ] of size n n where the (i, j) th entry in P is the probabiity of transitioning from node v i to node v j in one step, which is defined as foows p ij = { 1 deg(v i) v i v j 0 otherwise. (2) Let D = [d ij ] n n be a diagona matrix, where d ii = deg(v i ), then the transition probabiity is computed as P = D 1 A. 3.2 The Mixing Time Moving from a node to another on G is captured by the Markov chain which represents a random wak over G. A random wak of ength t over G is a sequence of vertices in G such that each node is seected at its predecessor in the random wak foowing the transition probabiity defined in (2). The Markov chain is said to be ergodic if it is irreducibe and aperiodic, where in such case it has a unique stationary distribution π and the distribution after t steps converges to π. The stationary distribution of the Markov chain is a probabiity distribution that is invariant to the transition matrix P (i.e., πp = π). The mixing time, T, is defined as the minima ength of the random wak to reach the stationary distribution. More precisey, the mixing time of a Markov chain on G parameterized by a variation distance ɛ is defined as T (ɛ) = max min{t : π π (i) P t 1 < ɛ}, (3) i 4

5 where π = [deg(v i )/2m] 1 n (for simpe graph), π (i) is the initia distribution concentrated at vertex v i, P t is the transition matrix after t steps, and 1 is the tota variation distance, defined as 1 n 2 j=1 π(i) j π j. We say that a Markov chain is fast-mixing [8, 23, 50, 51] when ɛ = Θ( 1 n ), and T (ɛ) = O(og n). An interesting resut on bounding the mixing time for simpe graphs is provided in [39], and used in [33] for measuring the mixing time of severa rea word socia graphs. In short, the method uses the second argest eigenvaue, µ for bounding the mixing time. The coser µ is to 1, the sower the mixing time, and faster otherwise. Measuring the mixing time can be done using both definitions [33]. However, since using the second argest eigenvaue woud account ony for the poorest mixing source in the socia graph, measuring the mixing time using the definition in (3) is highy desirabe to express the richer patterns of mixing, which represent various initia sources in the socia graph. In [33], we used this observation to capture the variety of mixing patterns in the same socia graph. By varying sources of waks, and samping such sources from G, one can get a good estimate about the distribution of the mixing across different sources, and thus the entire graph. By definition, however, the mixing time of the graph is defined as the mixing observed at the sowest source, thus every other source in the graph has at east that mixing bound. Tabe 1: Datasets of (undirected) socia graphs for which we measure the mixing time. Dataset # nodes # edges Youtube [28] 1, 134, 890 2, 987, 624 DBLP [24] 769, 641 3, 051, 127 Sashdot [22] 70, , 620 Epinion [38] 32, , 012 Physics 2 [20] 11, , 619 Gnutea [20] 4, , 742 Physics 1 [20] 4, , 422 Wiki-vote [19] 1, , Motivationa Initia Resuts We borrow the same toos in [33], which are expained section 3.2, to measure the mixing time of socia graphs and highight the variabiity in mixing patterns across different socia graphs. Datasets used for this measurement are shown in Tabe 1. For directed graphs (4 graphs in Tabe 3), we use the graph conversion method in section More detais on these graphs are in [33] and the corresponding citation aong with each graph. From each of these graphs, we seect 1, 000 initia nodes uniformy at random and compute the statistica distance ɛ as we increase the ength of the random wak t. At each time step we compute the basic statistics of ɛ the max, mean, and median. Resuts of this measurements are shown in Figure 1, where the main concusion [33] is that socia graphs differ in their mixing characteristics, independent of their size and density. This concusion can be made cear on Figure 1(b) (and on 1(a) and 1(c) for max and median with sight differences) by observing that whie Gnutea s number of nodes is 3 times arger arger than that of Wiki-vote, and whie Wiki-vote is 3 times denser than Gnutea, Gnutea sti has better mixing characteristics than any other graph we tested. Another exampe is by observing Youtube and Physics 2. Physics 2 dataset is three times denser and 100 times smaer, yet it has same mixing characteristics as Youtube. Other exampes can be easiy driven from these resuts to demonstrate this tendency. An interesting question that rises now is: can we reate the mixing time in these graphs to their structure? What is obvious is that simpe characteristics, ike density and size, are not be sufficient to answer this question. Moreover, degree distribution, custering coefficient, and others are not sufficient to understand the mixing pattern as shown in [9]. It is intuitive to think that fast-mixing graphs shoud have inherit property in their structure, which captures their connectivity. The connectivity itsef is reated to the mixing time. For exampe, the conductance which is an indicator of how we-connected is a graph gives a ower bound of the connectivity and mixing time. However, computing the optima conductance is NP-hard [2] and known approximation is a O(og n)-approximation which runs in O(n 2 ) [1], making it expensive to compute for arge graphs. Aso, even if the exact conductance is computed, it woud be used for reasoning about the ower bound of the mixing time, but not the genera pattern in the graph. Combining that with our findings in [33] where we show that some sow-mixing graphs according to the definition are sow due to a sma portion of nodes woud make the use of conductance to understand the mixing pattern in socia graphs of ess interest. We expore another graph structure property that coud sufficienty distinguish sow-mixing from fast-mixing graphs and show the pattern in their structure. More importanty, this property can be computed in O(n) for an exact soution, and can be done for arge (miion nodes) graphs feasiby. 4. THE MIXING AND GRAPH STRUCTURE It is uncear what properties sower mixing graphs possess. Furthermore, it has been caimed in the iterature that the mixing time do not reate to any of the graph structure properties, making the mixing time interesting in its own right [9]. Here, we re-examine this statement by trying to understand the mixing characteristics of socia graphs and reating them to graph structure. Indeed, we find that mixing characteristics 5

6 Epinion Wiki-vote Sashdot Gnutea Physics 1 Physics 2 DBLP Youtube Epinion Wiki-vote Sashdot Gnutea Physics 1 Physics 2 DBLP Youtube (a) Max (b) Mean (c) Median Figure 1: A measurement of the mixing time of undirected graphs shown in Tabe 1 [Best viewed in coors]. are very reated to core structure of the socia graph, which captures graph cohesiveness. We show that fast mixing graphs tend to have arge singe core, whereas sower mixing graphs tend to dissove into mutipe cores. Before getting into the detais of our measurements, we ay down the preiminaries by defining graphs cores. 4.1 Graph Cores and Node Coreness Let G = (V, E) be a graph defined according to the same mode in section 3. For any k where 1 k k max, et G k = (V k, E k ) be a subgraph in G such that V k = n k and E k = m k, and with the constraint that for a v i V, the minimum degree of any node v j V k is k. G k is said to be a k core of G if, in addition to the above condition, it is a maxima and connected graph. If we reax the connectivity condition of the k core, we get a set of cores (potentiay more than one) where each of such cores satisfy the degree condition. Such set of cores is represented as a (disconnected) graph G k = (V k, E k ) where V k = n k and E k = m k. An efficient agorithm for decomposing a simpe graph to its k cores by iterativey pruning nodes with degree ess than k has the compexity of O(m) where m is the number of edges in the graph [5]. We define the node-reative (or normaized) size of G k as n k /n and the edge-reative size of G k as m k /m. Simiary, for G k (the argest core) we define these normaized quantities as n k /n and m k /m. Finay, for every node v i V, we define the coreness c as the argest c s.t. v i G c where G c is a c core. The definition of k-core and graph coreness [25] is equivaent to many other interesting graph properties, such as k-cooring [10]. In particuar, a k-core graph is (k + 1)-coorabe, which does not ony depend on the density of the graph but the way edges are estabished between nodes. Indeed, we show in section 5 that increasing the density of the graph arbitrariy does not necessariy improve the mixing time. 4.2 Measurements and Resuts Epinion Wiki-vote Sashdot Gnutea Physics 1 Physics 2 DBLP Youtube We use the same datasets shown in Tabe 1 to expore the pattern of mixing time in socia graphs by decomposing them to their core structures defined above. For each of these graphs we use an off-the-shef impementation of the inear-time agorithm in [5] to compute the k core by reaxing the connectivity assumption, which coud ead to a disconnected graph. As k increases to its utimate vaue at which the graph disappears (no node in it woud have a c-coreness equa to that utimate k) we compute the foowing: (1) the number of cores in the k-core as we decompose the graph for increasing k, (2) the size of each k-core as we increase k normaized by the origina graph size, and (3) the size of each core in the k-core as we increase k. Among these we are in particuar interested in the first and second metrics the number and normaized tota size of the cores in the k-core. The resuts of these measurements shown in Figure 2 and Figure 3. Notice that graphs in Figure 2 are sow mixing and graphs in Figure 3 are fast mixing, as shown in Figure 1. We make two observations on these resuts that capture their fundamenta differences. First, the most obvious resut by comparing Figure 2 to Figure 3 is that sow mixing graphs are ess cohesive whereas fast mixing graphs are more cohesive. This cohesiveness is refected on the number of cores in the k-core of each graph as we increase k. Whereas sow mixing graphs shown in Figure 2 are decomposed into mutipe cores as we increase k, fast mixing graphs resist this decomposition and remain cohesive as we increase k (shown in Figure 3). Second, despite that sow mixing graphs are decomposed into mutipe cores, these cores are reativey sma in size and the disappearance of the graph is very quick in many cases as we increase k; see for exampe Figure 2(a)) and Figure 2(c). On the other hand, though they have singe core for an increasing k, they remain mosty arge in size and resist dissoution; see for exampe Figure 3(a) and Figure 3(c). Whie the genera tendency in the decrease of the normaized graph size as we increase k is reciproca in the order of k, with the difference being a constant among graphs, we observe some of the fact mixing have a negative exponentia order of k (e.g., c 1 e k + c 2 for two positive constants c 1 6

7 Number of cores # of cores graph size k (minimum degree) Graph size (normaized) Number of cores # of cores graph size k (minimum degree) (a) DBLP (b) Physics 1 (c) Youtube Figure 2: A measurement of the core structure in a seect of socia networks, representing both the number of cores in the k-core and the tota normaized size as k increases. Number of cores 1 0 # of cores graph size k (minimum degree) Graph size (normaized) Number of cores 1 0 # of cores graph size k (minimum degree) Graph size (normaized) Graph size (normaized) Number of cores Number of cores # of cores graph size k (minimum degree) # of cores graph size k (minimum degree) metrics we have considered here. Further theoretica and empirica investigation on this issue woud be an open direction to consider in the future. Now that we estabished that there is a fundamenta difference between fast and sow mixing graphs in terms of their structure, can we use these differences for sow mixing graphs to improve their mixing characteristics? We try to answer this question in the foowing section. 5. IMPROVING THE MIXING TIME Now that we earned the fundamenta difference in structure between fast and sower mixing socia graphs, a natura question that rises is how to improve the mixing time of sower mixing socia graphs. Indeed, there has been severa successfu attempts in the iterature to improve the mixing characteristics of Markov chains on graphs [3, 11, 14, 15, 35, 39]. A we-known exampe of such attempts is using teeportation probabiity [3] (which borrows ideas from [35]). Each node in the graph decides to foow the uniform random wak protoco by seecting the next hop of the wak from its direct neighbors uniformy at random, with a tota probabiity 1 α, (a) Wiki-vote (b) Sashdot (c) Epinion Figure 3: A measurement of the core structure in a seect of socia networks, representing both the number of cores in the k-core and the tota normaized size as k increases. and c 2 ). This atter case is iustrated in Figure 3(a) or direct the random wak towards a uniformy chosen where the decreases in the graph size is very sma. node at random from the rest of the graph with a tota Limitations of the measurements: it is obvious probabiity α. α is chosen to be sma enough so as not that our measurements ony consider two extremes: fast to deviate or change the overa random wak structure mixing graphs and sow mixing graphs but nothing in and the fina bounding or stationary distribution. between. In reaity, graphs tend to have mixing patterns that can be differentiated at average (or at maximum as 5.1 Limitations of Existing Soutions per the definition of the mixing time). These differences Whie such scheme in theory woud work and provide are not captured here by the method we considered, the intended guarantees in improving the mixing time, though we anticipate that this can be reated to the two it comes at high cost in practice [3]. Imagine an appication ike a Sybi defense that utiizes socia graphs for its bootstrapping, or an anonymous communication system that uses socia inks, or information dissemination agorithm that forwards information over socia inks, among many others. In each of these appications, users are imited by the number of nodes with whom they share reationships, and if such teeportation process existed it woud be with cost [3], which woud be high in many cases. Furthermore, incentives may not exist for using such scheme at scae. Another fundamenta reason why such an agorithm for improving the mixing time woud not be practica is that it improves the mixing time of nodes regardess to their abes: honest or dishonest. This in fact is due to that a node woud not be abe to verify whether a far away node that forwards the random wak towards it through those jumps is honest or not, since most ikey both nodes woud be separated by mutipe hops. In a Sybi defense that uses socia networks, whie one woud be interested in improving the mixing time of the honest region of the socia graph, Sybi (dishonest) nodes in the graph aso woud benefit from the teeportation probabiity used for performing the uniform jumps in Graph size (normaized) Graph size (normaized) 7

8 Origina X-1-C X-A-C X-A-A (a) Max Origina X-1-C X-A-C X-A-A (b) Mean Origina X-1-C X-A-C X-A-A (c) Standard Deviation Figure 4: Mixing time measurement of Physics 1 before/after using the heuristics to improve its mixing characteristics. Origina X-1-C X-A-C X-A-A (a) Max Origina X-1-C X-A-C X-A-A (b) Mean Origina X-1-C X-A-C X-A-A (c) Standard Deviation Figure 5: Mixing time measurement of Physics 2 before/after using the heuristics to improve its mixing characteristics. the random wak, thus vioating the basic assumption of sparse-cut used in the defense for identifying dishonest nodes. To this end, we outine two requirements for an idea agorithm that improves the mixing time in order to be sufficient for this type of appications. First, such an agorithm shoud improve the mixing time of the good nodes in the socia graph. Second, the same agorithm shoud maintain the property of the socia graph which differentiates honest from dishonest nodes. As in above, the agorithm shoud not improve the extent to which random waks originated from the dishonest region mix with the honest part of the graph. 5.2 Heuristics to Improve the Mixing Time To achieve both goas, using the structura property of sow-mixing socia graphs, we propose severa heuristics. In the k core graph decomposition process, a graph G i is decomposed into its G i+1 core by recursivey trimming a of its nodes with a degree ess than or equa to i. By doing so, we have observed that the graph G i+1 is decomposed into mutipe cores; one of these cores is the major core (argest in size), whie other cores are reativey smaer in size (minor cores). Whie it might be impractica, especiay for smaer i, to postpone or prevent the decomposition of the graph, one coud try severa techniques, which can be turned into agorithms to postpone or prevent the decomposition of the graph into mutipe disconnected components. One direct approach is to connect (or wire) disconnected cores in the graph G i to each other using auxiiary edges. This indeed eads to the foowing three scenarios. Scenario X-1-C: wire a singe node in each minor core X with a node in the major core C using one edge. The end vertices of added edges can be arbitrariy chosen. By doing so, we can easiy see that the graph wi aways have a singe core at any time when increasing k. The number of added edges is the sum of the number of cores in each k-core, for a k, minus k itsef (to excude major and singe cores). Scenario X-A-C: wire each node in each of the minor cores with a node in the major component, as we increase k. Same as above, this woud prevent producing mutipe cores at time and the number of auxiiary edges is bounded by the number of nodes in the minor component. Scenario X-A-A: wire a nodes in a minor core to other cores in the graph, incuding both minor and major cores. The number of auxiiary edges is bounded by the order of the number of nodes in each k-core. Notice that in each of these scenarios, we need to improve the mixing time of the honest region of the graph, so wiring a fixed number of honest nodes is ideay performed using this method. In our study, and for the imitations of our datasets, we consider that the entire graph is of honest nodes to derive insight on the potentia of each method in improving the mixing time. Extending this to the genera case when abes of nodes are given is straightforward. Furthermore, when considering wiring unabeed nodes even if an edge is cre- 8

9 ated between an attacker and an honest node, it woud have a imited impact on the number of Sybi identities introduced per that attack edge, as in SybiLimit for exampe. The rationae of using auxiiary inks is very natura and simpe. Such inks can be either virtuay created between nodes if the appication that uses the socia graph is centraized [46], or through ink recommendations if this process is to be performed in a decentraized appication [50]. 5.3 Resuts and Discussion We seect two of the sow-mixing and reativey sma socia graphs to expore the potentia of our heuristics. These graphs are shown in Tabe 1 with their initia statistics. The resuts of measuring the mixing time after appying the heuristics in section 5.2 and for the same settings of measurements as in section 3 are shown in Figure 4 and Figure 5, respectivey. The tota number of edges before and after wiring graphs according to the methods in section 5.2 is shown in Tabe 2. On these graphs we make foowing observations. First of a, our simpest heuristic (X-1-C), which produces minima effect on the graph density 122 edges are added to Physics 1 significanty improves the mixing time according to its definition as the maxima t for a given ɛ and provides a correction on the sowest mixing sources. This source, by definition, woud have the poorest mixing characteristics resuting in a poor mixing in the overa graph as shown in Figure 4(a). Simiar improvement is obtained on average for mixing time, as shown in Figure 4(b). The standard deviation among a source for which the mixing characteristics are computed is aso being improved as the addition of these few auxiiary edges reguate the structure of the graph, as shown in Figure 4(c). Second, the extent to which additiona edges improve the mixing time differs and depends on the initia mixing characteristics of the graph. Graphs that mix better than others (yet overa sow-mixing) tend to have ess number of cores which means ess number of additiona usefu edges are added to these graphs. For exampe, X-1-C adds about 68 edges to the origina Physics 2 graph, which transates into ess effect on the mixing time, as shown in Figure 5. The origina graph aready mixes better than Physics 1 dataset (on average) and the addition of these edges, though improves the sowest mixing sources (and the mixing of the graph as per the definition), it does not improve a ot on average. For the same reason, better improvement on average is reaized in Physics 1 as shown in Figure 4. Third, by considering the number of added edges in X-A-A in both datasets and the measured mixing time, we observe that the addition of a ot of edges despite improving the density of the graph does not improve Tabe 2: The number of edges in the different socia graphs before and after wiring them according to our heuristic for improving their mixing characteristics. The number of nodes is shown in Tabe 1. Dataset Number of edges (tota) Orig. X-1-C X-A-C X-A-A Physics Physics the mixing time. Indeed, we find that using this heuristic woud sow the mixing of these graphs down. One potentia reason for this behavior is that, as we add more edges, random waks are diverted from traversing other nodes in the graph by the additiona edges connecting minor cores. This diversion is transated into sower convergence to the stationary distribution. 6. THE IMPACT OF LINK DIRECTION Many graphs in genera, and socia graphs in particuar, are directed by nature. On the other hand, many appications, incuding Sybi defenses, information routing, dissemination, and anonymous communication when buit by expoiting socia structures require mutua reationships which produce undirected graphs. However, when undirected graphs are used as testing toos for these appications to bring insight on their usabiity and potentia, directed graphs are transformed into undirected graphs by omitting directions. One caim made in that context is the percent of added edges to compete edges in both directions and make them undirected is sma (e.g., 10% of the tota number of edges among a nodes), and thus converting the entire directed graph to undirected woud not incur a major impact on the graph structure. Whether this is the case or not when it comes to the mixing time of the socia graph is not tested before, though the intuition is that directed socia graphs woud have different, and ikey sower, mixing patterns than undirected graphs. Here, we investigate this issue and quaitativey measure the difference in the mixing time in both cases. Before introducing the measurements, we eaborate on the way to measure the mixing time in directed graphs and how this differs from the undirected case. 6.1 Mixing Time of Directed Socia Graphs Conditions required for defining the mixing time on undirected and directed socia graphs, and the way used for computing the mixing time, are sighty different. Whie connectivity is required in both cases to define the mixing time, the directed graph requires this connectivity in the form of a strongy connected component (SCC). In SCC there exists a path between every pair of nodes in the graph. Computing the SCC of a graph in 9

10 inear time can be done using Tarjan s agorithm, which we use in this work for different graphs. Notice that obtaining the SCC from the directed graph is necessary for defining the mixing time. From an appication point of view, if a design ike a Sybi defense is to be used on top of a directed graph then either that graph needs to be SCC at the first pace or to have a fixation for the isoated components in that graph so the entire graph is an SCC. One potentia way for fixing this probem to graphs that are not SCC in their entirety is to use our mixing time improvement method by wiring honest nodes in two disconnected components so as to aow fows in the entire graph. We eave this direction as a future work, and imit oursef to the SCC in these graphs which coud potentiay not incude a nodes. Whie the definition of the mixing time in both directed and undirected graphs is sti the same, the bounding (stationary) distribution of Markov chains defined on both graphs is different. In particuar, unike the undirected graph where we can easiy express the stationary distribution in a cean form in terms of node degree as shown in section 3, the stationary distribution in case of the directed graph does not reate to node degree (not the in-degree nor the out-degree) Defining the Mixing Time for Directed Graphs Simiar to the mode in section 3, et A = [a ij ] n n be the adjacency matrix of a directed graph G where a ij = 1 if an (directed) edge exists from node v i to node v j (denoted by v i v j ), and 0 otherwise. Notice that A is most ikey asymmetric, unike in undirected graphs. Let deg(v i ) be the out-degree of node v i. We define the transition probabiity matrix P = [p ij ] where p ij = 1/deg(v i ) iff v i v j and 0 otherwise. In a cean matrix form, we get P = (D ) 1 A, where D is a diagona matrix with the ii-th eement in it defined as j a ij. We define the stationary distribution as π = JP t (which coud be used for any connected graph), where t and J is a (1 n)-vector in which a entries are ones. As t grows, one woud ose the sparsity of P. Thus, computing the stationary distribution using the method above is time and space inefficient Estimating the Stationary Distribution Here we devise a method for computing a good estimate of the stationary distribution without having to modify the structure of P which eads to inefficiency. We observe that, regardess of the initia distribution, every wak on a strongy connected graph woud utimatey converge to the stationary distribution π. We coud further make the tota variation distance between the idea stationary distribution and the accumuated distribution of a random wak beginning from an arbitrary node in the graph arbitrariy sma. In other words, given an arbitrary initia distribution π ( ) and the matrix P, we can compute π ( ) = π ( ) P so as π ( ) π 1 < δ where δ is very cose to 0, for some arge wak ength. The convergence of π ( ) is guaranteed by the definition of the stationary distribution. Without knowing the (idea) stationary distribution π, one coud use π ( ) as an estimate for π, for arge. Such distribution is sufficient to measure a very cose estimate of the mixing time of the directed graph in an efficient way, and to serve the purpose of our measurement in understanding the difference between the mixing patterns in directed and undirected graphs under simiar circumstances. In [32] we bound the error between these measurements and idea measurements to a sma negigibe constant. Furthermore, to speed up the convergence to π, we begin from a uniform distribution π ( ) = [1/n] 1 n. Notice that computing π ( ) = π ( ) P does not require computing P. We can iterativey compute π ( ) using a vector-matrix mutipications of π ( ) and P, where we can benefit from the sparsity of P. In our measurements, and to make π ( ) a good estimate of π we set = 10, 000, for which we observe a steady distribution with no significant difference between π ( ) 1 and π( ) as increases Computing the Mixing Time The remaining part of computing the mixing time of directed graphs is straightforward. Given π ( ) and P, we define the mixing time as foows (simiar to Eq. 3, see Appendix A in [32] for bounding the error in this definition). T (ɛ) = max min{t : π ( ) i π (i) P t 1 < ɛ}, Using the same method as when computing the mixing time of the undirected graph, we begin by a sampe of initia distributions (of nodes) uniformy seected at random and compute ɛ as above for an increasing t. The mixing time as by definition is the maximum t for a given ɛ, whie other statistics such as the mean and standard deviation are good indicators of the genera tendency of the mixing pattern. 6.2 Resuts and Discussion Here we outine the main resuts and findings of measuring the mixing time of directed graphs. We compare our findings to the origina method in which directed graphs are converted into undirected graphs. This method is previousy used in [33] for measuring the mixing time, and foows the same method of graph converging as in [23] and [50] Datasets and Data Preprocessing Four different datasets are used for this measurement and are shown in Tabe 3. Three of these datasets are 10

11 Tabe 3: Datasets used for measuring the mixing time in directed graphs with their statistics. The undirected graph is obtained from the SCC and the difference is computed between 2m of undirected graph and m in directed graph. Dataset Origina graph Largest SCC Largest SCC % Undirected (SCC) Difference n m # SCC n m n (%) m (%) n m # edges percent Sashdot 77, , 468 6, , , , , , Epinion 75, , , , , , , , Wiki-vote 7, , 689 5, 816 1, , , , , Gnutea 10, , 994 6, 560 4, , , , , are shown in Tabe 3. Whie the same graph size is maintained, the difference in the number of edges and graph density between both forms of the graph is great, and ranges from as ow as 11% in Sashdot (agreeing with previous resuts on other datasets such as Livejourna experimented with in [23]) to as high as 50% as in Gnutea and Wiki-vote Resuts and Discussion Now we proceed to measure the mixing time for both directed and undirected graphs. Same as in section 3, we use the definition of the mixing time, with the stationary distribution propery computed for the different graphs based on their types. For each graph, we measure ɛ the distance between the stationary distribution and the accumuated distribution after t steps. We repeat this process for each graph, and by beginning from previousy used for measuring the mixing time in [33], 1000 different nodes as sources of initia distributions in and represent socia networks. The ast dataset (Gnutea) [21] order to capture the pattern of mixing. is of a peer-to-peer system for fie sharing, where nodes are hosts and edges indicate that a host is connected to another host. The first dataset is of the Sashdot Zoo [22], in which a directed edge between two nodes indicates that the The resuts of the measurements are shown in Figure 6 and Figure 7. Figure 6 shows a comparison between the average ɛ among 1000 sources for varying t from 1 to 100, whereas Figure 7 shows maximum ɛ for a given t among the 1000 sources, making t the mixing first node tags the second node as a friend. The second time for that ɛ by definition. Contrary to what dataset is of Epinion [38], a who-trust-whom onine socia network. An edge between two nodes indicates that the first node has tagged the second node as a trusted is anticipated due to the difference in the graph structure introduced by omitting edge direction, we find that the average mixing time does change in most cases, as node. The third dataset is of wiki-vote [19], which contains voting for wikipedia administrators promotion. A ink between two nodes indicates that the first node has voted for the second node. For each of these graphs, in order to satisfy the connectivity condition required for measuring the mixing time, we compute the argest strongy connected component (SCC). The SCC of each graph and its reative size compared to the origina graph is shown in Tabe 3. The argest SCC varies in size, and ranges from as ow as 18% of the number of nodes in the origina graph (as ow as 38% of edges, as in Wiki-vote) to as high as 90% of nodes (and edges, as in Sashdot). We convert each of the SCC s computed above from the directed graph form to an undirected graph by competing the in- and out-degrees, and omit oops to obtain simpe graphs. The resuting graphs and their statistics shown in figures 6(a) through 6(c), which are representative socia graphs. More interesting, we find that the mixing time of the directed graph is even better than that of the undirected graph, as it is the case in Figure 6(d). Whie the mean computed over a ɛ s for a given t is meaningfu for the average node, it does not capture the worst case scenario which is of interest in many cases. For exampe, for an anonymous communication systems suggested on top of socia networks, it is aways better to prove guarantees with ower-bounds. Lower bound guarantees are satisfied by the mixing time in the definition, and satisfied for the maximum ɛ among a distributions for a given t. For the same experiment above, we pot maximum ɛ for different t, where the resuts are shown in Figure 7. As the per the definition of the mixing time, no difference between measurements for both types of graphs is observed (in Figure 7(b) and Figure 7(c)). Furthermore, the same advantage of mixing time pointed out earier for Gnutea dataset is aso transated into a better mixing time when considering the maximum among a sources, as shown in Figure 7(d). Finay, whie the average mixing characteristics of Epinion shown in Figure 6(a) are tied for both directed and undirected graphs, the maximum ɛ computed among a sources for a given t differs sighty, as shown in Figure 7(a). Furthermore, in the same figure, we observe that a steep region of the convergence to the stationary distribution ends at some point, which happens earier in the directed graph case. Even before switching the convergence rate, both graphs provide simiar mixing characteristics, though the directed graph mixes better in the active region (for t < 20). 11