A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article we utilize inherent network symmetry to collapse all redundant information from a network, resulting in a coarse-graining which we show to carry the essential structural information of the 'parent' network. In the context of algebraic combinatorics, this coarse-graining is known as the quotient. We systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of 'real-world' quotients with respect to those of their parent networks. In particular, we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity (heterogeneity and hubs vertices) and communication (diameter and mean geodesic distance), suggesting that quotients constitute the essential structural skeleton of their parent network. We summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms.  Many physical systems -from the world-wide web to scientific collaborations and biochemical reactions inside cells -can be modeled as networks. The ubiquity of empirical networks has generated increasing interest in their study over the last decade during which much progress has been made toward elucidating general network organizational principles beyond the specific details of individual systems [1, 2, 3, 4, 5, 6] . Structural properties which are commonly found in many disparate networks include: the 'small-world' property [5] ; the scale-free distribution of vertex degrees[6]; hierarchical modularity [7] ; network construction from motifs[8]; assortative mixing [9] ; and self-similarity[10] amongst others. Together, investigation of generic structural properties such as these may be thought of as an attempt to understand network complexity [11] . In order to find simplicity in this complexity some authors have attempted to extract network 'skeletons': related networks which capture essential structural features of the system from which they are derived, but are simpler in some quantitative way. Existing network skeletons include for instance, the fractal skeleton [12] , which is responsible for fractal scaling; and the communication skeleton [13] , which is responsible for the majority of communication flow through the network. Such skeletons are generally formed with respect to a given property, for example fractality or communication, and thus do not represent a structural skeleton in the strongest sense. In this article we propose an alternative skeleton which formally captures all essential structural information, and which can be significantly smaller than the original network from which it is derived. The method we use is based upon utilizing inherent network symmetry. Although almost all large random networks are asymmetric [14] , many empirical real-world networks are surprisingly richly symmetric [15, 16, 17, 18] . This symmetry commonly results from the presence of locally treelike or biclique-like structures [16, 18] which are present in many empirical networks, and derive naturally from elementary growth processes such as growth with preferential attachment [16] and growth with similar linkage pattern [18] . However, despite a rich abstract theory of graph symmetry [19, 20, 21, 22] , the symmetry structure of complex real-world networks has not yet been explored extensively. Intuitively, a network is symmetric if two or more of its vertices can be permuted without altering vertex adjacency. Symmetric networks therefore necessarily contain a certain degree structural redundancy in that they possess multiple vertices which play the same structural role. Thus network symmetry is strongly related to network redundancy.In this paper we use this relationship to 2 show how symmetry also provides a natural means to formally exclude redundancy while still preserving essential network structure, by factoring out structurally identical elements. The structure of the remainder of this paper is as follows: first we introduce essential background material concerning network automorphism groups, and show how a network's automorphism group may be used to produce a coarse-grained skeleton of the network called the quotient. We also introduce a variation on the classical quotient which we call the s-quotient. Then we show that quotients can be substantially smaller than the network from which they are derived and explore ways in which key structural properties are inherited by the quotient and the s-quotient from the 'parent' network. In particular, we shall examine how network heterogeneity, degree distribution and communication properties are carried from the parent to its quotients. BACKGROUND AND DEFINITIONS
If we collapse all structural redundancy characterized by network symmetry, we can obtain a structural skeleton of the parent network -network quotient, which preserves various key functional properties of the parent network REF .
Network Quotients: Structural Skeletons of Complex Systems
Physical Review E, Vol 78,Page: 046102, 2008
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