We study the problem of graph partitioning, or clustering, in sparse networks with prior information about the clusters. Specifically, we assume that for a fraction ρ of the nodes the true cluster assignments are known in advance. This can be understood as a semi-supervised version of clustering, in contrast to unsupervised clustering where the only available information is the graph structure. In the unsupervised case, it is known that there is a threshold of the inter-cluster connectivity beyond which clusters cannot be detected. Here we study the impact of the prior information on the detection threshold, and show that even minute [but generic] values of ρ > 0 shift the threshold downwards to its lowest possible value. For weighted graphs we show that a small semi-supervising can be used for a non-trivial definition of communities. Graph partitioning is an important problem with a wide range of applications in circuit design, social sciences, data mining, etc. In the context of social network analysis, this problem is better known as community detection, where community is loosely defined as a group of nodes so that the link density within the group is higher than across different groups. Many real-world networks have well-manifested community structure [1] , and much recent research has focused on developing community detection methods (see [2] for a review). A common observed feature of many of those methods is the presence of a threshold in inter-community coupling beyond which communities cannot be detected [3, 4] . This problem has been studied by formulating the community detection as a minimization of a certain Potts-Ising Hamiltonian [4, 5] . It has been shown that that the graph partitioning problem is indeed characterized by a phase transition from detectable to undetectable regimes as one increases the coupling strength between the clusters [4] . Most work so far has considered unsupervised version of clustering, where the only available information is the graph structure. In many situations, however, one might have additional information about possible cluster assignments of certain nodes. This version of the problem has attracted recent interest in the machine learning community, in the context of graph-based semi-supervised learning and classification (see [6] for a recent survey). In this setting, one assumes that, besides the graph structure, one also has some prior information about the cluster-membership of a set of nodes [7, 8] . Our purpose is to present a theoretical analysis of the semi-supervised version of the community detection problem for simple, bi-community networks, and to uncover new scenarios of community detection facilitated by semi-supervising. Model. Consider an Erdös-Rényi graph: each pair of nodes is linked with probability α/N , where N is the number of nodes in the graphs. Each link is given a weight J > 0. Now imagine a pair of such identical Erdös-Rényi graph, which models two clusters (communities). Besides the intra-cluster J-links, each node in one graph is linked with probability γ/N with nodes in the other graph. Each of these inter-cluster links is given a weight K > 0. For clarity, both J and K are assumed to be integer numbers. This planted bisection graph model will be employed for studying the performance of the cluster detection method, which places an Ising spin on each node and lets these spins interact via the network links [4, 5] : where we have made the bi-cluster nature of the network explicit by introducing separate spin variables s i = ±1 ands i = ±1 (i = 1, . . . , N ) for two clusters. Here J ij and J ij are identically and independently distributed random variables which assume zero with probability 1 − α N and J > 0 with probability α N . Likewise, K ij identically and independently are equal to zero with probability 1 − γ N and to K > 0 with probability γ N . To enforce equipartition, the Hamiltonian (1) will be studied under the constraint Thus, detecting the sign of a given spin s i at zero temperature (so as to exclude all thermal fluctuations) we can conclude to which cluster the corresponding node belongs: all spins having equal signs belong to the same cluster. The error probability p e for the cluster assignment is where m is the (single-cluster) magnetization, . . . T =0 is the zero-temperature Gibbsian average, i.e. the average over all configurations of spins having in the thermodynamic limit the minimal energy given by (1, 2), and where [. . .] av is the average over the bi-graph structure, i.e., over {J ij }, {J ij } and {K ij }. As implied by the self-averaging feature, instead of taking [. . .] av , we can evaluate s i T =0 on the most probable bi-graph structure(s).
Allahverdyan et al. REF studied the problem of semi-supervised graph clustering by integrating known cluster assignments for a fraction of nodes.
Community Detection with and without Prior Information
Europhysics Letters 90, p.18002, 2010
Computer Science Physics Biology