Structural Balance in Networks. An Optimizational Approach. Andrej Mrvar. Faculty of Social Sciences. University of Ljubljana. Kardeljeva pl.

Transcription

1 Structural Balance in Networks An Optiizational Approach Andrej Mrvar Faculty of Social Sciences University of Ljubljana Kardeljeva pl Ljubljana March

2 Contents 1 Balanced and clusterable signed graphs Signed graphs : : : : : : : : : : : : : : : : : : : : : : : : : : : Soe properties of balanced and clusterable signed graphs : : 2 2 Optiizational approach to the proble of balanced and clusterable graphs Local optiization : : : : : : : : : : : : : : : : : : : : : : : : Solving the proble of balance using local optiization : : : : 5 3 Ipleentation of progra for solving proble of balance in signed graphs Ipleentation of local optiization procedure : : : : : : : : Representation of signed graphs : : : : : : : : : : : : : : : : : Exaples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : How to use progra CLUSTER : : : : : : : : : : : : : : : : : 12

3 1 Balanced and clusterable signed graphs 1.1 Signed graphs A signed graph is an ordered pair (G; ) where [1]: G = (V; R) is a graph without loops with set of vertices V arcs R V V ; and set of : R! fp; ng is a sign function. The arcs with the sign p are positive while the arcs with the sign n are negative. The proble of balance can be forulated in the following way: Is it possible to cluster vertices of a signed graph into two or ore groups, so that every positive relation occurs between vertices in the sae group, and every negative relation occurs between vertices in dierent groups? A signed graph is called balanced if all vertices can be clustered into exactly two groups. If all vertices can be clustered into ore than two groups the graph is called clusterable. Exaple fro [3]: A group of people can be represented by a signed graph, where vertices correspond to the individuals; a positive edge joins two vertices if there is a positive relation between the corresponding individuals and a negative edge joins two vertices if there is a negative relation between the corresponding individuals. Indierence between two individuals is indicated by the lack of any edge joining the corresponding vertices. If this group of people represents a balanced syste, then the group can be divided into two subgroups, so that there will be no eneies within the subgroup, and there will be no friends in dierent subgroups. Soe social psychologists [3] have hypothesized a tendency toward balance in any social syste, iplying that an unbalanced syste contains excessive stress and tension. 1.2 Soe properties of balanced and clusterable signed graphs A path in a signed graph is called positive if it has an even nuber of negative edges and is called negative otherwise. [3] 2

4 Soe theores which give necessary and sucient conditions for a signed graph to be balanced (or clusterable) can be found in [1], [2], [3], [4], [6]. For exaple: Theore 1 A signed graph is balanced i for every two vertices in graph all paths joining the have the sae sign. Theore 2 A signed graph is balanced i every closed seiwalk is positive. Two exaples will help us to understand the iportance of these two theores (exaples fro [3] and [6]): Consider a group of people again. Pairs of individuals are friends or eneies. Suppose anyone would tell the ruour which has only two basic fors, one true and one false. Everyone in this syste will tell the ruour to a friend in the sae for he had received it, but would change the for if he were to pass the ruour to his eney. If the syste is balanced, each person will hear only one version of the ruour. Also the person who started the ruour will hear it returned to hi in the sae for as he originally knew it. Signed graphs that represent relations between people can also be used in analysis of literature. Most of novels have the following structure: Relations between people are balanced in the beginning. Afterwards the situation becoes very coplicated the graph that represents it is unbalanced and in the end happy end is expected. But if we exaine all paths joining two vertices in our unbalanced graph, we can predict who will be friends and who eneies in the end of the story. If ore than two clusters are tolerated the second theore will be a little dierent: Theore 3 A signed graph is clusterable i it contains no closed seiwalk with exactly one negative arc. Using these theores an algorith can be constructed to verify if a signed graph is clusterable: we ust just exaine signs of all paths and signs of all closed seiwalks in that graph. 3

5 But we want ore. We want to nd an optial solution (clusters in a signed graph) if it exists. If there is no optial solution we want to nd a solution which has the least errors. What are actually errors of a given solution? An error is every negative link between two vertices in the sae cluster and every positive link between two vertices which belong to dierent clusters. These ideas can be used in an optiizational approach to the proble of balance of signed graphs. 2 Optiizational approach to the proble of balanced and clusterable graphs 2.1 Local optiization Local optiization is a procedure, which enables us to nd a iniu of a function. The only diculty is that we cannot distinguish between local and global iniu. But if we repeat this procedure several ties we can coe very close to global iniu or we can even nd the global iniu. How does local optiization procedure work in general? [5] Local optiization procedure is deterined using following objects: Set of all feasible solutions Criterion function P :! R which tells us how good is a particular solution. Neighbourhood relation, which deterines which solutions can be obtained fro a given solution in one step of local optiization (which solutions are neighbours of a given solution). We start a local optiization procedure with a rando initial feasible solution. Then we exaine all neighbours of a given solution. If we nd a solution with saller value of criterion function we ove to that solution. The process is nished when there is no better solution in a neighbourhood of a given solution. This solution is then called a local iniu. 4

6 2.2 Solving the proble of balance using local optiization If we use local optiization to solve the proble of balance of signed graphs the above objects are: Set of all feasible solutions is the set of all clusterings with a given nuber of clusters. Criterion function is nuber of errors nuber of negative links between two vertices in the sae cluster plus nuber of positive links between two vertices which belong to dierent clusters. Neighbours of a given clustering are clusterings which can be obtained fro a given clustering by: oving one vertex fro one to another cluster interchanging two vertices fro dierent clusters 3 Ipleentation of progra for solving proble of balance in signed graphs 3.1 Ipleentation of local optiization procedure Local optiization procedure will give us good solutions if all decisions which are not known are ade randoly. So we start procedure with a rando clustering into given nuber of clusters. Criterion function is coputed for this clustering. Then we exaine neighbours of a given solution and copute values of criterion function for that neighbours. If better solution is found we ove to this solution. Two facts are iportant in selecting neighbours: We entioned above that neighbours can be obtained by oving a vertex fro one cluster to another or by interchanging two vertices fro dierent clusters. The ethod of neighbours deterination is randoly selected. If the local transforation is always selected it the sae order the process is not so ecient. There are two dierent ways to continue depending on selected local transforation: 5

7 If interchanges are chosen we do not exaine all possible interchanges ordered nuerically or alphanuerically, but the list of all possible interchanges is randoly generated. If ovings are chosen the rando perutation of vertices and rando perutation of clusters are ade and these perutations deterine the ovings order. Epty clusters are not allowed. When we copute value of criterion function in a selected neighbour we do not need to copute criterion function for the whole graph, what would give us tie coplexity of O(n 2 ). Instead of this we just have to copute the dierence in criterion function, which is caused by a oveent or interchange. In this case the tie coplexity is only O(n), what is uch better. 3.2 Representation of signed graphs The following input les are recognized by the progra: 1. Words that begin with * are reserved they ust be present in input le. % eans coent: Everything following that sign will be ignored by the progra. Vertices are dened using *Vertex stateent. First, nuber of vertices is dened, then all vertices are described in the following way: vertex nuber, vertex nae, (x, y) coordinates in the plane (x and y are nubers between 0 and 1) and radiu of a circle that represents the vertex. Then directed and undirected links are dened using *Arcs and *Edges stateent. *Arcs for exaple stands for: Vertex 10 is connected with vertex 11 with negative arc. (1 eans positive link, 1 eans negative link). Clustering which is obtained using the progra is appended at the end of the le as properties (*Properties) of this graph. Exaple: *Network Saple6 %Network fro Roberts p. 76 a *Vertices 11 6

8 1 a b c d e f g h i j k % *Arcs *Edges % *Properties a, b, c, e, h, j}, d, f, g, i, k} } 0 a, b, c, e, h}, d, g, i, k}, f, j} } 0 This graph is presented in gure 1. 7

9 2. The dierence between the rst and the second type of input le is only in representation of links which can be also dened using a atrix for (*Matrix). Exaple: *Network Saple2 % Batagelj page 6 *Vertices *Matrix % *Properties 1, 5, 6, 9}, 2, 3, 4}, 7, 8} } 0 This graph is presented in gure This is the shortest representation of a signed graph. Nubers in each line ean: Vertex nuber, x and y coordinates of vertex (between 0 and 1) and list of all neighbours of the given vertex. If one connection 8

10 is negative then the neighbour has a negative sign. Exaple: This graph is presented in gure Exaples Look now at gures of graphs which were entioned in three exaples above: Siilar gures are also used in progra (positive links are green lines, negative links are yellow dotted lines). We will also consider clusterings for these exaples. Graph in gure 1 has clusterings with 2, 3, 4, 5, 6 in 7 clusters without errors. Exaples of these clusterings: a, b, c, e, h, j}, d, f, g, i, k} } a, b, c, e, h}, d, g, i, k}, f, j} } a, b, c, e, h}, d}, f, g, j}, i, k} } a, b, c, e}, d}, f, g, j}, h}, i, k} } a, b, c, e}, d}, f}, g, j}, h}, i, k} } a, b, c, e}, d}, f}, g}, h}, i, k}, j} } 9

11 g b 6 / a c e - h HY 6 H H H H? - H k j i f? Figure 1: Directed graph Graph in gure 2 is undirected all links are in both directions, so the direction is not arked. This graph is not balanced, but we can nd soe clusterings with 2 clusters which have 4 errors: 1, 2, 3, 4, 5, 6, 9}, 7, 8} } 1, 5, 7, 8}, 2, 3, 4, 6, 9} } 1, 5, 6, 7, 9}, 2, 3, 4, 8} } Graph can be partitioned into 3 clusters without errors: 1, 5, 6, 9}, 2, 3, 4}, 7, 8} } The best clustering with 4 clusters has 2 errors: 10

12 Figure 2: Undirected graph 1, 5, 6, 9}, 2, 3}, 4}, 7, 8} } 1, 5, 6, 9}, 2}, 3, 4}, 7, 8} } Clusterings with ore than 4 clusters have uch ore errors. Graph in gure 3 can be partitioned in several ways: Graph is balanced: 1, 2, 3, 4, 5, 6}, 7, 8, 9, 10, 11, 12} } Clustering with 3 clusters is optial too: 1, 2, 3, 4, 5, 6}, 7, 9, 11}, 8, 10, 12} } 11

13 Figure 3: Undirected graph 3.4 How to use progra CLUSTER Progra Cluster is ipleentation of local optiization approach to the proble of balanced and clusterable signed graphs. It is written in Turbo Pascal 6.0. Here are soe basic instructions how to use the progra. To run the progra you should type for exaple: cluster saple g a b c d e f Six paraeters (a, b, c, d, e, f) are: a) Nae of the progra. b) Nae of the input le with graph description (one of three possible types as dened above). The default extension is.net. If you have soe other extension you should type it (for exaple Network.dat). Instead of using only sign ( or ) to represent relationship, also integer 12

14 values can be used to express the strength of relationship (friendship for exaple). Larger values represent stronger relationship. These values are also used when criterion function is coputed. c) Nuber of clusters you want to divide your graph to. (for exaple 2 or 3 or 4,...) d) Nuber of repetitions of local optiization procedure (fro randoly selected initial partition to local iniu). The best solution fro all repetitions is selected. For ore coplicated graphs ore repetitions are needed to nd a good solution. If the optial solution (solution without errors) is found before the last repetition is nished the progra is stopped. e) Factor that tells us the ratio between iportance of negative and positive errors: T otal:error = (neg:errors) (1? ) (pos:errors) 0 < 0:5 positive errors are ore iportant than negative = 0:5 positive and negative errors are equally iportant 0:5 < 1 negative errors are ore iportant than positive > 1 all links are considered as positive < 0 all links are considered as negative f) If you type the last paraeter 'g' the progra will run in graphical ode (VGA is needed). If the last paraeter is not 'g' non-graphical representation will be used. The local optiization procedure starts with randoly selected initial partition. But you can also suggest the progra which initial partition should be used (for exaple you get this partition using soe other approach and you want to optiize it). You can do that by using another input le with the sae nae as le with description of the graph and extension.inp. For exaple if you run the progra as written above (nae of the graph is saple6.net) and the le saple6.inp is found on the directory the initial partition will be that one which is recoended in le saple6.inp. This input le should look for the graph with 11 verteces in gure 1 like this: 13

15 So the nuber of lines ust be equal to the nuber of clusters in which we want to cluster our graph (in this exaple 2). Values in the sae line correspond to the vertices' nubers that belong to the sae initial cluster. If graphical representation is used the resulting clusters are displayed so that vertices that correspond to the sae cluster are coloured with the sae colour. If the solution is not optial, the wrong relations are displayed with thick lines. If type 1 or type 2 input les are used the resulting clustering is written as properties in the sae (input) le together with nuber of errors and value of. The result is written only in case that this solution is found the rst tie (it is not written yet as properties). If type 3 input le is used the result is written to the le with the sae nae as input le and extension.out. References [1] Batagelj V.: Seirings for Social Networks Analysis [2] Batagelj V.: Ovojnica vrednostne atrike grafa, Obzornik at. z. 37 (1990) 4 [3] Chartrand G.: Introductory Graph Theory. Chapter 8: Graphs and Social Psychology p Dover Publications, Inc., New York (1985) [4] Davis J. A.: Clustering and Structural Balance in Graphs. Huan Relations 20(1967), p [5] Ferligoj A.: Razvrscanje v skupine. Metodoloski zvezki 4, Ljubljana 1984 [6] Roberts F. S.: Discrete Matheatical Models. (With Applications to Social, Biological and Environental Probles) Chapter 3: Signed Graphs and the Theory of Structural Balance p (1976) Prentice Hall, Inc., Englewood Clis, New Yersey 14